 Weak type with a normed domain

For a normed space $E$ and $0<q<\infty$, an operator $T:E\to L^0$ is of weak type $(E,q)$ when the displayed distribution bound holds uniformly over $f\in E$ and $a>0$. No completeness of $E$ is needed. A sublinear operator has pointwise triangle domination and absolute homogeneity; its outputs may be understood modulo almost everywhere equality.